MASTER'S THESIS Particle Tracking in Circular Accelerators Using the Exact Hamiltonian in SixTrack Mattias Fjellström 2013 Master of Science in Engineering Technology Engineering Physics and Electrical Engineering Luleå University of Technology Department of Engineering Sciences and Mathematics Particle tracking in circular accelerators using the exact Hamiltonian in SixTrack Mattias Fjellstr¨om [email protected] Dep. of Engineering Sciences and Mathematics Lule˚aUniversity of Technology Supervisors: Riccardo De Maria (CERN) and Johan Hansson (LTU) December 2013 ii Abstract Particle motion in accelerators is in general complex. Tracking codes are developed to simulate beam dynamics in accelerators. SixTrack is a long lived particle tracking code maintained at CERN, the European Organization for Nuclear Research. A particle accelerator consists of a large number of magnets and other electromagnetic devices that guide the particle through the accelerator. Each device defines its own equation of motion, which often cannot be solved exactly. For this purpose, a number of approximations are introduced in order to facilitate the solution and to speed up the computation. In a high-energy accelerator, the particle has small transverse momentum components. This is exploited in the small-angle approximation. In this approximation the equations of motion are expanded to a low order in the transverse momentum components. In low-energy particle accelerators, or in tracking with large momentum deviations, this approximation is invalid. The equations of motion of a particle passing through a field-free region in an accelerator, a so called drift space, has been implemented in the SixTrack code. The equations of motion are derived from the exact Hamiltonian, keeping the non-linear term unexpanded. This solution of the drift is called the exact drift space. Previously, the drift space has been solved using the small-angle approximation. This solution of the drift is called the expanded drift space. The new implementation is a step towards a more realistic, and more general, tracking code. The drift space contains the bulk of the small-angle approximation in a tracking code, it is therefore the most important element to address. The new drift space implementation is applied in two simulation studies on the Large Hadron Collider (LHC). In the first, particle losses in the collimation system of the machine are studied. The collimation system is a collection of protective devices, used to protect the rest of the accel- erator from particles spiraling out of the machine. The application of the exact drift space in this simulation shows a small, but insignificant, variation compared to the expanded drift. Of the total 14 × 106 tracked particles, about 12 × 106 are absorbed in the collimators for each model. The total number of particles lost in other locations of the ring are about 12 × 103 for both models. The most dangerous losses are losses in the superconducting magnets, called cold losses. For the exact drift, the number of cold losses were 4471. This is a short increase from the expanded drift, where the number of cold losses were 4446. These results do not show that the exact drift space is necessary in collimation studies for the LHC. It should still be an improvement to consider for future machine protection studies. The second simulation study on the LHC is an investigation of the tune variation as a function of the momentum deviation of the particle. The tune is a measure of the number of oscillations a particle makes during one complete turn around the accelerator. The number of oscillations must avoid certain values to not induce a resonance in the motion, causing the motion to be unstable. The momentum deviation, δ, is a measure of the momentum of a particle compared to an ideal reference particle. The horizontal{ and vertical tunes were calculated for a range of values for δ, both with the exact{ and expanded drift space. As expected the deviation between the models grows with a larger momentum deviation. The maximum differences in the simulation were obtained for δ = −4 × 10−3, where the exact model results in a tune value larger by 3 × 10−5 for the horizontal tune and 1:5 × 10−5 for the vertical tune. These tune shifts are small, and for regular tracking simulations in the LHC they are insignificant. However, in simulations where very high-order resonance effects are considered, these tune shifts could start to become important. iii iv Preface This Master of Science project has been carried out as a Technical Student in the BE-ABP-LCU (Beams department, Accelerators and Beams Physics, LHC Commissioning and Upgrade) section at CERN (the European Organization for Nuclear Research) in Geneva, Switzerland, under the supervision of Riccardo de Maria. I would like to thank all the people who have helped me with this work. Most of all I wish to thank my supervisor at CERN, Riccardo, for his guidance in this project and for introducing me to the world of accelerator physics and also for recommending me to use Linux - which will be a permanent tool in the rest of my career. I would also like to thank my supervisor at LTU, professor Johan Hansson, for proof reading this manuscript and for offering some of the most interesting physics courses at LTU, without which I might not have ended up at CERN. Thanks also to Pascal Hermes, now working on his PhD at CERN, for the help with the collimation simulations and for providing the tools needed for analyzing the results. This work concludes my years as a student in Engineering Physics and Electrical Engineering with specialization in Computational Methods and Physics at LTU. Mattias Fjellstrom, Geneva, December 2013 v vi Contents Abstract iii Preface v 1 Introduction and motivation 1 1.1 The CERN accelerator complex . .1 1.2 Particle tracking . .4 1.3 Problem formulation . .4 1.4 Existing solutions . .5 1.5 Aim and purpose . .5 1.6 Method . .5 1.7 Thesis structure . .6 2 Theoretical background 7 2.1 Coordinate system . .7 2.2 The accelerator Hamiltonian . .8 2.2.1 Hamiltonian mechanics . .8 2.2.2 Phase space . .9 2.2.3 Canonical transformation . .9 2.2.4 Particle motion in an accelerator . 10 2.3 Accelerator physics . 11 2.3.1 Components of an accelerator . 12 2.3.2 Transverse dynamics . 13 2.3.3 Dispersion . 14 2.3.4 Longitudinal dynamics . 14 2.4 Symplectic integration . 15 2.4.1 The symplectic condition . 15 2.4.2 Example of symplectic integration . 16 2.4.3 Symplectic integration in particle tracking . 17 2.5 Exact Hamiltonian . 18 2.5.1 The exact drift space . 18 2.5.2 The expanded drift space . 19 3 SixTrack 21 3.1 Purpose of a particle tracking code . 21 3.2 Code structure and building SixTrack . 22 3.3 Structure of a SixTrack simulation . 23 3.4 Input to SixTrack . 23 3.4.1 FREE/GEOM . 24 3.4.2 Initial values for tracking (INIT) . 24 3.4.3 Tracking parameters (TRAC) . 25 3.4.4 Synchrotron oscillations (SYNC) . 25 vii viii CONTENTS 3.4.5 Single elements (SING) . 26 3.4.6 Block input (BLOC) . 26 3.4.7 Structure of elements (STRU) . 27 3.5 Output from SixTrack . 27 4 Implementation and benchmarking 29 4.1 Implementation details . 29 4.1.1 Flag for exact tracking . 29 4.1.2 Tracking routines . 30 4.1.3 Differential algebra closed orbit and optics calculations . 32 4.2 Benchmark codes . 34 4.2.1 MAD-X . 34 4.2.2 PTC . 35 4.2.3 Relation to SixTrack . 35 4.3 Benchmark in 4D and 6D . 36 4.3.1 Lattice and settings . 36 4.3.2 Results . 37 4.4 Large transverse momentum . 38 4.4.1 Lattice and settings . 38 4.4.2 Results . 38 4.5 Computational speed . 39 4.5.1 Lattice and settings . 39 4.5.2 Results . 39 5 Application 41 5.1 Collimation . 41 5.1.1 LHC Collimation system . 41 5.1.2 Collimation extension to SixTrack . 42 5.1.3 Application of drift in collimation routines . 42 5.1.4 Simulation of losses . 43 5.1.5 Comparison and results . 43 5.2 Tune shift in the LHC . 45 5.2.1 Resonances . 45 5.2.2 Simulation of tune shifts . 45 6 Discussion and conclusions 47 6.1 Benchmark results . 47 6.2 Impact on collimation . 47 6.3 Impact on tune variation . 48 6.4 Further simulations . 48 6.5 Comment on SixTrack . 48 Bibliography 51 A Common acronyms 55 B Derivation of the accelerator Hamiltonian 57 B.1 Straight coordinate system . 57 B.2 Curved coordinate system . 59 C Exact dipole implementation 61 C.1 Equation of motion . 61 C.2 Implementation details . 62 C.2.1 Tracking routines . ..